# Is every T$_1$-space a finite-closed topological space?

My general topology textbook introduced T$$_1$$-spaces as the following:

A topological space $$(X,\tau)$$ is said to be a T$$_1$$-space if:

• $$\forall x \in X$$, the set $$\{x\}$$ is closed

They also stated the following proposition:

If $$(X,\tau)$$ is a topological space, then:

(1) $$X$$ and $$\emptyset$$ are closed sets

(2) The intersection of any (finite or infinite) number of closed sets is closed

(3) The union of a finite number of close sets is also closed

Now, let $$A \subseteq X$$. We have that $$A= \bigcup_{x \in A} \{x\}$$. If $$A$$ is finite, then $$A$$ is the union of a finite number of sets (3), thus every finite subset is closed. If $$A$$ is infinite, then this is the union of a infinite number of closed sets, hence it's not closed. Doesn't this make $$\tau$$ the finite-closed topology?

• The union of an infinite number of closed sets may or may not be closed in general. Consider the usual topology on $\mathbb R$ Jul 20 '20 at 16:07
• The discrete topology on any set is $T_1$, and in it every set is closed. In particular, if we give $\Bbb R$ the discrete topology, then every union of closed sets is closed, no matter how many sets are involved. Jul 20 '20 at 18:38

"If $$A$$ is infinite, then this is the union of a infinite number of closed sets, hence it's not closed"
In general, $$p\to q$$ is not the same as $$\sim p\to \sim q$$.
There are $$T_1$$ space which does not have finite-closed topology. $$\mathbb R$$ with the standard topology is one of the example.
Every metric space is $$T_1$$ and an infinite union of closed sets is not a non closed set necessarily. For example $$\bigcup\limits_{x\in \mathbb{R}}\{x\}$$ is closed since it is the whole space or conider any other infinite closed set, $$F$$, of $$\mathbb{R}$$ for example; then $$F=\bigcup\limits_{x\in F}\{x\}$$ is closed.